Do the mentioned acts - such as repeating certain supplications in specific ways, commanding to repeat "from the evil of the whisperer, the slinker" 300 times after each prayer, and reciting specific surahs on water for bathing with it and disposing of it in a pure place - make this Ruqyah doubtful as a legitimate Islamic Ruqyah?

What was mentioned does not contain unknown incantations (ruqya) or polytheistic expressions (shirk (definition)). Some scholars have permitted the recitation of ruqya over water, which is then drunk by the afflicted person, as Imam Ahmad did. It has been authentically reported from the Salaf that they would recite (Qur'an) over water or similar substances, and then drink it or bathe with it, because the words of Allah Almighty are a healing. The same applies to reciting over oil or food, and then drinking it, anointing oneself with it, or bathing with it.